Nonlinear Integro-differential Equations Research Articles

Numerical Solution of Non-linear Integro-differential Equations using Operational Matrix based on the Hosoya Polynomial of a Path Graph

Objectives: Introduction to new numerical techniques to solve differential, difference, and integro-differential equations (IDEs) are always remaining the thrust area of research for many scientists over the centuries. The prime objective of this work is to contribute a new numerical technique to solve IDEs. Method: To address non-linear integro-differential equations, we computed an operational matrix of derivatives based on the Hosoya polynomial of the path graph in this work. Findings: Using the derived operational matrix, we have solved both Volterra and Fredholm integrodifferential equations. Taking suitable examples accuracy of the projected method is demonstrated in this paper in terms of a graphical representation of the absolute error. The results of the examples reveal that the projected method is a suitable method to solve IDEs. Novelty: The application of the Hosoya polynomial of path graph to solve integro-differential equations is a novel approach in the field of numerical analysis. Keywords: Volterra Integrodifferential equations; Fredhlom Integrodifferential equations; Graph theorypolynomials

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modeled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi, and by Pazy and Rabinowitz.

Existence and Stability Results for Nonlinear Implicit Random Fractional Integro-Differential Equations

Existence and Stability Results for Nonlinear Implicit Random Fractional Integro-Differential Equations

Haar Collocations Method for Nonlinear Variable Order Fractional Integro-Differential Equations

Haar Collocations Method for Nonlinear Variable Order Fractional Integro-Differential Equations

New algorithm to solve nonlinear functional equations applying linearization then double discretization scheme (L.D.D)

Solving functional nonlinear equations leads to the question: which to begin with, linearization or discretization? Recent papers confirmed that linearizing then discretizing (L.D) is better. In this paper, we develop a new numerical scheme that begins with the linearization phase and then, a double discretizations process (L.D.D). This new method gives a different theoretical framework where the convergence process is satisfied under some hypotheses. Many examples are offered to show the effectiveness of our new scheme. Starting with a comparison of the obtained numerical results with the results of a recently published research, moreover, applications to the system of nonlinear integro-differential equations. Obtained numerical results show that our (L.D.D) method is more efficient for solving nonlinear functional equations.

Oscillating Behavior of a Compartmental Model with Retarded Noisy Dynamic Infection Rate

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected (I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) to an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes.

Exact Solutions of One Nonlinear Countable-Dimensional System of Integro-Differential Equations

In the present paper, a nonlinear countable-dimensional system of integrodifferential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a doublesided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system’s solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.

Methods of numerical analysis for some integral dynamical systems with delay arguments

The aim of this work is to construct direct and iterative numerical methods for solving functional equations with hereditary components. Such equations are a convenient tool for modeling dynamical systems. In particular, they are used in population models structured by age with a finite life span. Models based on integro-differential and integral equations with various kinds of delay arguments are considered. For nonlinear equations, the operators are linearized according to the modified Newton-Kantorovich scheme. Direct quadrature and simple iteration methods are used to discretize linear equations. These methods are constructed in the paper: an iterative method for solving a nonlinear integro-differential equation on the semiaxis (−∞,0] , a direct method for solving the signal recovery problem, and iterative methods for solving a nonlinear Volterra integral equation with a constant delay. Special quadrature formulas based on orthogonal Lagger polynomials are used to approximate improper integrals on the semiaxis. The results of numerical experiments confirm the convergence of suggested methods. The proposed approaches can also be applied to other classes of nonlinear equations with delays.

An Iterative Scheme for Solving Arbitrary-Order Nonlinear Volterra Integro-Differential Equations Involving Delay

This paper introduces an iterative-based numerical scheme for solving nonlinear fractional-order Volterra integro-differential equations involving delay. Additionally, we provide sufficient conditions for the existence and uniqueness of the solution. The composite trapezoidal rule is applied to approximate the integral involved in the equation, followed by discretizing the Caputo fractional derivative operator of arbitrary order $$\alpha \in (0,1)$$ by using the classical L1 scheme. Further, the Daftardar-Gejji and Jafari method is employed to solve the implicit algebraic equation. The convergence analysis and error bounds of the proposed scheme are presented. It is shown that the approximate solution converges to the exact solution with order $$( 2-\alpha ).$$ We illustrate the efficacy and applicability of the proposed method through a couple of examples.

Uniqueness of a nonlinear integro-differential equation with nonlocal boundary condition and variable coefficients

This paper studies the uniqueness of solutions to a two-term nonlinear fractional integro-differential equation with nonlocal boundary condition and variable coefficients based on the Mittag-Leffler function, Babenko’s approach, and Banach’s contractive principle. An example is also provided to illustrate the applications of our theorem.

Projection Methods for Neural Field Equations

Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields and derive a priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice depends on the choice of the projector. Using an elementary time stepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector and one from the time discretization. We give examples of concrete projection methods: two collocation schemes (piecewise linear and spectral collocation) and two Galerkin schemes (finite element and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests.

A Nonlinear Schrеdinger Equation for Gravity-Capillary Waves on Deep Water with Constant Vorticit

The surface gravity-capillary waves on deep water with constant vorticity in the regionbounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrödinger equation is derived from a system of exact nonlinear integro-differential equations in conformal variables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrödinger equation, the role of the mean flow is taken into account. The nonlinear Schrödinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrödinger equation that represents a soliton of the “ninth wave” type is obtained. bounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrödinger equation is derived from a system of exact nonlinear integro-differential equations in conformalvariables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrödinger equation, the role of the mean flow is taken into account. The nonlinear Schrödinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrödinger equation that represents a soliton of the “ninth wave” type is obtained.

Analytical-based exact-kernel vibration and long-term creep stress and large deformation redistributions of the suddenly pressurized incompressible visco-hyperelastic thick cylinders

The long-term creep and short-term damped dynamic redistributions of the large deformations and stresses are investigated here for long, suddenly pressurized, rubber-like visco-hyperelastic thick-walled cylinders, for the first time. The extension of the Mooney–Rivlin strain energy density function to incompressible visco-hyperelastic materials is accomplished by using a hereditary integral based on the Prony series and an exponential relaxation kernel. To extract a more accurate/efficient formulation, the relaxation kernel is processed instead of the common approximate discretization of the time derivative of the stress tensor in the convolution integral. The resulting nonlinear integrodifferential equations whose number of terms grows with time are solved by a novel analytical approach in conjunction with the second-order Runge–Kutta time-marching, bound-varying trapezoidal technique, and an iterative/updating scheme. In comparison to the commercial finite element analysis codes, the present analytical formulation and solution algorithm lead to higher accuracies from the mathematical point of view. The visco-hyperelastic properties are extracted from previously reported experimental results, using an error minimization fitting technique. Results reveal that in contrast to the hyperelastic vessel, the oscillations of the displacements and stresses of the visco-hyperelastic cylinder occur around levels that grow with time. Moreover, smaller relaxation parameters lead to smaller displacements and stresses and the responses intend earlier to the steady-state condition for higher values of the stiffness elements of the Prony series. In long term, the mean stresses and mean radial displacements of the inner and outer layers of the cylinder grow asymptotically with time but the largest overshoots and amplitudes of the responses occur at the early times. However, smaller values of the relaxation parameter expedite the convergence to the steady-state condition.

Asymptotic stability of nonlinear neutral delay integro-differential equations

Abstract In this paper, by using Sadovskii’s fixed point theorem and the properties of the measure of noncompactness, we establish some sufficient conditions for the asymptotic stability results of nonlinear neutral integro-differential equations with variable delays. The results presented in this paper improve and generalize some results in the literature. An example is considered to illustrate our main results.

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors.

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Abstract In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.

An Investigation on Existence, Uniqueness, and Approximate Solutions for Two-Dimensional Nonlinear Fractional Integro-Differential Equations

In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials via the collocation method to reduce the equations into a system of equations. Error bounds of the presented method are obtained. Five test problems are solved. The obtained numerical results show the accuracy, efficiency, and applicability of the proposed approach.

Postcritical Behavior of Nonlocal Strain Gradient Arches: Formulation and Differential Quadrature Solution

Arches are important components in nanostructures and systems. Because of the remarkable strength of nanomaterials, nanoarches are slender, thus enhancing their vulnerability to geometric instability, such as buckling. Although molecular simulations are often employed to analyze nanostructures, their use in routine analysis/design is formidable due to prohibitively exhaustive computation. Equivalent continuum models were developed as alternatives. The buckling and postcritical phenomena of the classical arch also form an important benchmark problem in nonlinear mechanics. This study investigates the buckling and the postcritical behavior of nanoarch subjected to inward pressure using nonlocal (NL), strain gradient (SG) continuum theory. The governing equations are derived in the form of a sixth-order nonlinear integrodifferential equation, unlike the fourth-order for a classical arch. The equation is then solved numerically using a differential quadrature (DQ) with appropriate boundary conditions. An incremental-iterative arc-length continuation is employed for the solution of the resulting algebraic system of equations. The equilibrium paths are traced for the possible instability modes, such as symmetric/antisymmetric bifurcations, snap-through and limit point instability, ascertained with the internal-external force diagrams. A particular instability mode is triggered at a certain threshold/range of the slenderness ratio of the arch, which is significantly influenced by the NL and SG interactions. These interactions not only cause quantitative changes in the instability behavior but also lead to qualitative changes, such as cessation, shift, and conversion of modes, more prominently for SG arches. Similar to classical arches, the prebuckling nonlinearity is shown to be significant.

ОБ ОЦЕНКЕ СНИЗУ РЕШЕНИЙ СЛАБО НЕЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА С ЗАПАЗДЫВАНИЯМИ

Sufficient conditions are established for lower bounds and tending to infinity as the argument of solutions to a weakly non-linear second-order Volterra integro-differential equation with delays on the half-axis grows unbounded. For this purpose, a non-standard method of reduction to a system, the method of transforming the equations of V. Volterra, the method of partial cutting, the method of integral inequalities with delays and the method of estimating solutions from below Yu.A. Ved are developed. It is shown that solutions with initial data from a certain manifold have lower bounds, and such solutions tend to infinity according to the exponential law with unlimited growth of the argument. This article proposes a technique for solving the problem of estimating from below on the half-line solutions of a weakly nonlinear Volterra-type integro-differential equation of the second order with delays, while the main role is played by the non-standard method of reduction to the system and the method of estimating solutions from below Yu.A.Ved (1965).

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in $$L^2$$ -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm.